In this section we had a review on explicit and recursive formulas. We found that while using an explicit formula we only need the first term and the common difference in an arithmetic sequence. Then in a recursive we need the first term and previous term which is a bit inconvenient when you are trying to find the 100th term you have to do the sequence about 100 times to get to the answer. So if you want to find a term of a sequence that is very large like 300 it would be best to use the explicit formula.
-allison r.
A Geometric Sequence is one in which the ratio of consecutive terms is consistent. The ratio can be a fraction or a whole number.
THEOREM:
Let n be a positive integer and g1 and r be constants. The formulas
gn = g1r^(n-1) and g1
gn = rg n-1, n>1
generate the terms of the geometric sequence with first term g1 and constant ratio r. The first is explicit and the second is recursive.
In lesson 8-2, we learned about limits. This is similar to what we learned in lesson 6-4, the limiting value of the sequence (1+1/n)^n which is named "e".
A limit of a sequence can be identified as a sequence that gets infinitely closer to a particular x value (asymptote) as n increases but never actually touches that number. A description of what happens to the values Tn of a sequence as n gets very large is known as the END BEHAVIOR. A sequence which doesn't have a finite limit is said to be DIVERGENT or TO DIVERGE. A sequence which has a finite limit L is said to be CONVERGENT or to CONVERGE TO L. One example of a convergent sequence is a HARMONIC SEQUENCE. A harmonic sequence is the sequence of reciprocals of the positive integers; 1, 1/2, 1/3, 1/4 etc. with a limit of 0. Graphically, the terms of the harmonic sequence lie on one branch of the hyperbola Y=1/n. There is also another well-known convergent sequence known as THE ALTERNATING HARMONIC SEQUENCE. The alternating harmonic sequence with terms Cn=((-1)^n) / n is just like the harmonic sequence, but every other term is negative. As the harmonic sequence it's limit is 0 and has the horizontal asymptote of Y=0. In general, there are 7 properties that limits follow. To study the limits, go to the section 8-2 slide show and they will be included in the slides. By using the properties of limits ,you can find out if a particular sequence actually does have a limit.
-austin r.
Limit Properties
(1) The limit of the harmonic sequence is 0.
(2) The limit of a constant sequence is that constant.
(3 & 4) The limit of the sum or product is the sum or product of the individual limits
(5) The limit of a constant times the terms of a sequence equals the product of the constant and the limt of the sequence.
(6) The limit of a reciprocal is the reciprocal of the limit.
(7) The limit of a quotient is the quotient of the individual limits. austin a.
A series is the sum of a number of terms of a sequence. An infinite series is a series with an infinite number of terms. A finite series is a series with a set number of terms. An arithmetic series is a series whose terms are taken from an arithmetic series. The sum of an arithmetic series with first term a1 and constant difference d is given by Sn = n/2 (a1 + an) or Sn = n/2 [2 a1 +(n-1)d].
-Dan H.
In lesson 8-3 we learned about Arithmetic Series remember the definitions but also you need to remember ( ) in example 1. a) there were ( ) around the equation so you take n^2 and then - 1( 1^2=1 1-1=0...8^2=64 64-1=63) then add all of these together. But in 1. b) notice that the ( ) are only aruond n^2 so you take 1^2=1...8^2=64 after that you add all of them up and only then do you subtra - 1. It is the same for 4. b) in the book 2(1)=2...2(5)=10 after you have all the answers you add them and only then do you + 1.
Geometric Series - an indicated sum of terms of a geometric sequence
Theorem: S(sub n) = (g(sub 1) * (1-(r^n)) ) / (1-r) pg. 510 in your book
This theorem will output the sum of the first n terms with first term g(sub 1) and constant ratio r.
Examples in your book can be found on pages 510 - 513
- Austin B.
A Geometric Series is a series whose terms are taken from a geometric series. Nth Partial Sum is the sum of the first n terms of a series.
for ex: Sn= g1+g1r+g1r^2+...+g1r^n-1. The theorem for this lesson is Sn=g1(1-r^n)/(1-r). When you multiply the numerator and denominator of this formula by -1, you get the formula for situations where r is > then 1. Sn = g1(r^n-1)/(r-1). Hope this helped.
Infinite Series- is the sum of a sequence with infinite values expressed infinity over sigma i=1 with the sum of a sub i=a sub 1+ a sub 2+ a sub 3+...
The sum of the Series, S sub infinity for an infinite series infinity over sigma i=1 sum of a sub i is the limit of the sequence of partial sums S sub n of the series, provided the limit exists and is finite, this can be written S sub n=n over sigma i=1 a sub i, so S sub infinity=limit S sub n= limit n over sigma i=1 a sub i.
Next is a theorem to summarize the work on finding the sum of an infinite geometric series!
g sub 1+ g sub 1 *r + g sub 1 * r^2 + g sub 1 * r^3+... + g sub 1 * r ^ n-1+... with g sub 1 not = 0
so if we deal with similar situations and patterns, we can find that if the absolute value of r < 1 the series converges and S sub infinity=g sub 1 divided by 1-r, and the second determination is that if the absolute value of r is > or equal to 1, the series diverges, or goes on forever. It was difficult understanding the lesson, however if anyone has any questions, watch the wikispace info provided or read through your book, IT WILL HELP TO CLEAR THINGS UP!
Troy K. Longenecker Professional in Sheep Castration and Artificial Insemination
Combinations
In this lesson you learn that a comb. is just like a perm. But here you count the numbers of ways that elements of ways that elements of a set may and can be arranged. Order is not important!
The notation is written as n C r = n! / (n - r)! r!
EX: 20 C 2 = 20 / (20 - 2)! 2! = 20! / 18! 2! = 20x19 / 2 = 190 units
Remember the arrangement and order is important!
Z Longenecker
OOOHHH YEA
Combinations are collections of objects in which the order doesn't matter.
The theorems of combinations and permutations are very similar... Combination (nCr=(n!/(n-r)!r!))-----Permutation (nCr=(n!/(n-r)!)
The difference between the two is that a permutation accounts for all possible events with the order mattering, as where a combination gives the number of possible events but eliminates things that occur the same twice, this is don by adding r! into the denominator of what would be a permutation.
N MYERS
In this lesson we learned that for any given value n, we could calculate nC0, nC1, nC2,.....nCn (a total of n+1 calculations). By looking more closely at the values of nCr systematically, we learned that a pattern developed. When these array of value are placed in the form of a right triangle it looks something like this:
row 0 -------> . 1
row 1 -------> . 1 1
row 2 -------> .1 2 1
row 3 -------> .1 3 3 1
row 4 -------> .1 4 6 4 1
.........and so on (page 529 in textbook)
This triangle is known in the Western world as Pascal's Triangle.
We also learned an important theorem in this lesson that states: If n and r are positive integers with r greater than or equal to n, then the (r+1)st term in row n of Pascal's Triangle is nCr.
Another important concept we learned is the 4 properties of Pascal's Triangle:
1. The first and last terms in each row are 1s. -----------> nC0 = nCn = 1*
2. The second and next-to-last terms in the nth row equal n. ------------> nC1 = nCn-1 = n*
3. Each row is symmetric. --------------> nCr=nCn-r*
4. The sum of the terms in row n is 2^n. -------------> n
. E nCr = 2^n*
. r = 0
*for any whole number n
----------------------------------------------------------------------------------------------------Zach G.-------------------------------------------------------------------------------------
In this lesson, we learned about Pascal's Triangle. Pascal's triangle is and array in the form of a right triangle. Each row begins with 1s. The sum of the terms in a row is 2^n. It is very easy to find the next row. All you have to do is add the two numbers above it to find the term you are looking for. Finding the terms this was is known as the symmetry property.
Mary G.
In this lesson, we learned about the Binomial Theorem, which states that for any nonnegative integer n, (x+y)^n = nC0 x^n + nC1 x^(n-1)y + nC2x^(n-2)y2 + ... + nCk x^(n-k)y^k + ... + nCn y^n. So it may look confusing, but the theorem is really quite simple. The combinations (i.e. nC0) are essentially following a row of Pascal's Triangle. The degree of the resulting polynomial corresponds to the appropriate row of Pascal's Triangle. For instance, If we were to expand (x + y)^3, we would know that the coefficients would be found in the 3rd row (3rd degree = 3rd row; nth degree = nth row). All that's left to do is working with the exponents themselves. The first exponent would be x^3, as seen in the problem. The following three would be x^2*y, x*y^2, and y^3. Notice that the variable alphabetically first (x) will have exponents that descend, and the following variable has exponents that ascend (y). So for x, the order of exponent values goes 3, 2, 1, 0; y is the opposite. Now we have to combine what we know. In the third row of Pascal's Triangle, the values are 1, 3, 3, 1. So, let's put what we know together. 1x^3 + 3x^2*y + 3x*y^2 + 1y^3. We can get rid of the ones, so the final expansion is x^3 + 3x^2*y + 3x*y^2 + y^3. It is important to remember that x and y are variables, so any number can replace them. 2g could replace x and 34u could replace y. In that case, the resulting expansion would not follow a row of Pascal's Triangle. Bear this in mind. Also, there are many uses of the Binomial Theorem, for instance in counting problems. Well, hopefully this entry helps!
-Bobby M.
We learned the binomial theorem in this lesson. The binomial theorem is - for any nonnegative integer n, (x+y)^n = nC0 x^n+nC1 x^(n-1)y + .... nCk x^(n-k)y^k + .... + nCn y^n. Pretty much what you do is follow pascal's triangle to figure these problems out. The number before C is what row of Pascal's Triangle you were working with and the number after is what term in the row you are dealing with. That's pretty much what you do for the problems.
Check out this link for a slide-by-slide explanation. You can leave an audio comment, too!
A binomial experiment needs to have repeated trials of a fixed number and two possible outcomes (a success [S] and a failure [F]). For example to find the probability of getting exactly two sums of five when tossing two fair dice 5 times you have two outcomes and a set number of repeated trials. The trials also need to be independent of each other and have the same probability. Binomial probability distribution is the probability that is generated from a binomial experiment. It normally deals with the probability of having a success. ~Katie Sholly~
In section 8-9 we learned about binomial experiments and other things of that nature. In the
binomial experiment there is a fixed number of trials with only two possible outcomes. (S= Success, F= Failure) The percentage of the Success and Failure added together must be equal 1. The variable n is used to represent the number of trials and the probability of success is p. -Diesel
Table of Contents
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Chapter 8: Sequences, Series, and Combinations
8-1: Formulas for Sequences
In this section we had a review on explicit and recursive formulas. We found that while using an explicit formula we only need the first term and the common difference in an arithmetic sequence. Then in a recursive we need the first term and previous term which is a bit inconvenient when you are trying to find the 100th term you have to do the sequence about 100 times to get to the answer. So if you want to find a term of a sequence that is very large like 300 it would be best to use the explicit formula.
-allison r.
A Geometric Sequence is one in which the ratio of consecutive terms is consistent. The ratio can be a fraction or a whole number.
THEOREM:
Let n be a positive integer and g1 and r be constants. The formulas
gn = g1r^(n-1) and g1
gn = rg n-1, n>1
generate the terms of the geometric sequence with first term g1 and constant ratio r. The first is explicit and the second is recursive.
!!!!!ANDREW CLARK!!!!!!!
8-2: Limits of Sequences
In lesson 8-2, we learned about limits. This is similar to what we learned in lesson 6-4, the limiting value of the sequence (1+1/n)^n which is named "e".
A limit of a sequence can be identified as a sequence that gets infinitely closer to a particular x value (asymptote) as n increases but never actually touches that number. A description of what happens to the values Tn of a sequence as n gets very large is known as the END BEHAVIOR. A sequence which doesn't have a finite limit is said to be DIVERGENT or TO DIVERGE. A sequence which has a finite limit L is said to be CONVERGENT or to CONVERGE TO L. One example of a convergent sequence is a HARMONIC SEQUENCE. A harmonic sequence is the sequence of reciprocals of the positive integers; 1, 1/2, 1/3, 1/4 etc. with a limit of 0. Graphically, the terms of the harmonic sequence lie on one branch of the hyperbola Y=1/n. There is also another well-known convergent sequence known as THE ALTERNATING HARMONIC SEQUENCE. The alternating harmonic sequence with terms Cn=((-1)^n) / n is just like the harmonic sequence, but every other term is negative. As the harmonic sequence it's limit is 0 and has the horizontal asymptote of Y=0. In general, there are 7 properties that limits follow. To study the limits, go to the section 8-2 slide show and they will be included in the slides. By using the properties of limits ,you can find out if a particular sequence actually does have a limit.
-austin r.
Limit Properties
(1) The limit of the harmonic sequence is 0.
(2) The limit of a constant sequence is that constant.
(3 & 4) The limit of the sum or product is the sum or product of the individual limits
(5) The limit of a constant times the terms of a sequence equals the product of the constant and the limt of the sequence.
(6) The limit of a reciprocal is the reciprocal of the limit.
(7) The limit of a quotient is the quotient of the individual limits. austin a.
8-3: Arithmetic Series
A series is the sum of a number of terms of a sequence. An infinite series is a series with an infinite number of terms. A finite series is a series with a set number of terms. An arithmetic series is a series whose terms are taken from an arithmetic series. The sum of an arithmetic series with first term a1 and constant difference d is given by Sn = n/2 (a1 + an) or Sn = n/2 [2 a1 +(n-1)d].
-Dan H.
In lesson 8-3 we learned about Arithmetic Series remember the definitions but also you need to remember ( ) in example 1. a) there were ( ) around the equation so you take n^2 and then - 1( 1^2=1 1-1=0...8^2=64 64-1=63) then add all of these together. But in 1. b) notice that the ( ) are only aruond n^2 so you take 1^2=1...8^2=64 after that you add all of them up and only then do you subtra - 1. It is the same for 4. b) in the book 2(1)=2...2(5)=10 after you have all the answers you add them and only then do you + 1.
Alyssa W.
8-4: Geometric Series
Geometric Series - an indicated sum of terms of a geometric sequence
Theorem: S(sub n) = (g(sub 1) * (1-(r^n)) ) / (1-r) pg. 510 in your book
This theorem will output the sum of the first n terms with first term g(sub 1) and constant ratio r.
Examples in your book can be found on pages 510 - 513
- Austin B.
A Geometric Series is a series whose terms are taken from a geometric series. Nth Partial Sum is the sum of the first n terms of a series.
for ex: Sn= g1+g1r+g1r^2+...+g1r^n-1. The theorem for this lesson is Sn=g1(1-r^n)/(1-r). When you multiply the numerator and denominator of this formula by -1, you get the formula for situations where r is > then 1. Sn = g1(r^n-1)/(r-1). Hope this helped.
Lantz M.
8-5: Infinite Series
Infinite Series- is the sum of a sequence with infinite values expressed infinity over sigma i=1 with the sum of a sub i=a sub 1+ a sub 2+ a sub 3+...
The sum of the Series, S sub infinity for an infinite series infinity over sigma i=1 sum of a sub i is the limit of the sequence of partial sums S sub n of the series, provided the limit exists and is finite, this can be written S sub n=n over sigma i=1 a sub i, so S sub infinity=limit S sub n= limit n over sigma i=1 a sub i.
Next is a theorem to summarize the work on finding the sum of an infinite geometric series!
g sub 1+ g sub 1 *r + g sub 1 * r^2 + g sub 1 * r^3+... + g sub 1 * r ^ n-1+... with g sub 1 not = 0
so if we deal with similar situations and patterns, we can find that if the absolute value of r < 1 the series converges and S sub infinity=g sub 1 divided by 1-r, and the second determination is that if the absolute value of r is > or equal to 1, the series diverges, or goes on forever. It was difficult understanding the lesson, however if anyone has any questions, watch the wikispace info provided or read through your book, IT WILL HELP TO CLEAR THINGS UP!
Troy K. Longenecker Professional in Sheep Castration and Artificial Insemination
8-6: Combinations
Combinations
In this lesson you learn that a comb. is just like a perm. But here you count the numbers of ways that elements of ways that elements of a set may and can be arranged. Order is not important!
The notation is written as n C r = n! / (n - r)! r!
EX: 20 C 2 = 20 / (20 - 2)! 2! = 20! / 18! 2! = 20x19 / 2 = 190 units
Remember the arrangement and order is important!
Z Longenecker
OOOHHH YEA
Combinations are collections of objects in which the order doesn't matter.
The theorems of combinations and permutations are very similar... Combination (nCr=(n!/(n-r)!r!))-----Permutation (nCr=(n!/(n-r)!)
The difference between the two is that a permutation accounts for all possible events with the order mattering, as where a combination gives the number of possible events but eliminates things that occur the same twice, this is don by adding r! into the denominator of what would be a permutation.
N MYERS
8-7: Pascal's Triangle
NLVM Interactive Applet
Shodor Applet 1
Shodor Applet 2
In this lesson we learned that for any given value n, we could calculate nC0, nC1, nC2,.....nCn (a total of n+1 calculations). By looking more closely at the values of nCr systematically, we learned that a pattern developed. When these array of value are placed in the form of a right triangle it looks something like this:
row 0 -------> . 1
row 1 -------> . 1 1
row 2 -------> .1 2 1
row 3 -------> .1 3 3 1
row 4 -------> .1 4 6 4 1
.........and so on (page 529 in textbook)
This triangle is known in the Western world as Pascal's Triangle.
We also learned an important theorem in this lesson that states: If n and r are positive integers with r greater than or equal to n, then the (r+1)st term in row n of Pascal's Triangle is nCr.
Another important concept we learned is the 4 properties of Pascal's Triangle:
1. The first and last terms in each row are 1s. -----------> nC0 = nCn = 1*
2. The second and next-to-last terms in the nth row equal n. ------------> nC1 = nCn-1 = n*
3. Each row is symmetric. --------------> nCr=nCn-r*
4. The sum of the terms in row n is 2^n. -------------> n
. E nCr = 2^n*
. r = 0
*for any whole number n
----------------------------------------------------------------------------------------------------Zach G.-------------------------------------------------------------------------------------
In this lesson, we learned about Pascal's Triangle. Pascal's triangle is and array in the form of a right triangle. Each row begins with 1s. The sum of the terms in a row is 2^n. It is very easy to find the next row. All you have to do is add the two numbers above it to find the term you are looking for. Finding the terms this was is known as the symmetry property.
Mary G.
8-8: The Binomial Theorem
In this lesson, we learned about the Binomial Theorem, which states that for any nonnegative integer n, (x+y)^n = nC0 x^n + nC1 x^(n-1)y + nC2x^(n-2)y2 + ... + nCk x^(n-k)y^k + ... + nCn y^n. So it may look confusing, but the theorem is really quite simple. The combinations (i.e. nC0) are essentially following a row of Pascal's Triangle. The degree of the resulting polynomial corresponds to the appropriate row of Pascal's Triangle. For instance, If we were to expand (x + y)^3, we would know that the coefficients would be found in the 3rd row (3rd degree = 3rd row; nth degree = nth row). All that's left to do is working with the exponents themselves. The first exponent would be x^3, as seen in the problem. The following three would be x^2*y, x*y^2, and y^3. Notice that the variable alphabetically first (x) will have exponents that descend, and the following variable has exponents that ascend (y). So for x, the order of exponent values goes 3, 2, 1, 0; y is the opposite. Now we have to combine what we know. In the third row of Pascal's Triangle, the values are 1, 3, 3, 1. So, let's put what we know together. 1x^3 + 3x^2*y + 3x*y^2 + 1y^3. We can get rid of the ones, so the final expansion is x^3 + 3x^2*y + 3x*y^2 + y^3. It is important to remember that x and y are variables, so any number can replace them. 2g could replace x and 34u could replace y. In that case, the resulting expansion would not follow a row of Pascal's Triangle. Bear this in mind. Also, there are many uses of the Binomial Theorem, for instance in counting problems. Well, hopefully this entry helps!
-Bobby M.
We learned the binomial theorem in this lesson. The binomial theorem is - for any nonnegative integer n, (x+y)^n = nC0 x^n+nC1 x^(n-1)y + .... nCk x^(n-k)y^k + .... + nCn y^n. Pretty much what you do is follow pascal's triangle to figure these problems out. The number before C is what row of Pascal's Triangle you were working with and the number after is what term in the row you are dealing with. That's pretty much what you do for the problems.
Travis D.
8-9: Binomial Probabilities
Download the slides (Error in loading to SlideShare):
Check out this link for a slide-by-slide explanation. You can leave an audio comment, too!
A binomial experiment needs to have repeated trials of a fixed number and two possible outcomes (a success [S] and a failure [F]). For example to find the probability of getting exactly two sums of five when tossing two fair dice 5 times you have two outcomes and a set number of repeated trials. The trials also need to be independent of each other and have the same probability. Binomial probability distribution is the probability that is generated from a binomial experiment. It normally deals with the probability of having a success.
~Katie Sholly~
In section 8-9 we learned about binomial experiments and other things of that nature. In the
binomial experiment there is a fixed number of trials with only two possible outcomes. (S= Success, F= Failure) The percentage of the Success and Failure added together must be equal 1. The variable n is used to represent the number of trials and the probability of success is p.
-Diesel
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